Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian

Abstract

We present a notion of weak solution for the Dirichlet problem driven by the fractional Laplacian, following the Stampacchia theory. Then, we study semilinear problems of the form (Equation Presented) when the nonlinearity f and the boundary data g, h are positive, but allowing the right-hand side to be both positive or negative and looking for solutions that blow up at the boundary. The operator E is a weighted limit to the boundary: for example, if Ω is the ball B, there exists a constant C(n, s) > 0 such that (Equation Presented) Our starting observation is the existence of s-harmonic functions which explode at the boundary: these will be used both as supersolutions in the case of negative right-hand side and as subsolutions in the positive case.